(dy)/(dx)+2y=7,y(0)=1

{ "query": { "display": "$$\\frac{dy}{dx}+2y=7,\\:y\\left(0\\right)=1$$", "symbolab_question": "ODE#\\frac{dy}{dx}+2y=7,y(0)=1" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "ODE", "subTopic": "FirstSeparable", "default": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{dy}{dx}+2y=7,\\:{\\quad}y\\left(0\\right)=1:{\\quad}y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}$$", "input": "\\frac{dy}{dx}+2y=7", "steps": [ { "type": "interim", "title": "Solve separable ODE:$${\\quad}y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}$$", "input": "\\frac{dy}{dx}+2y=7", "steps": [ { "type": "definition", "title": "First order separable Ordinary Differential Equation", "text": "A first order separable ODE has the form of $$N\\left(y\\right){\\cdot}y'=M\\left(x\\right)$$" }, { "type": "step", "primary": "Substitute $$\\frac{dy}{dx}$$ with $$y^{\\prime}\\left(x\\right)$$", "result": "y^{^{\\prime}}\\left(x\\right)+2y=7" }, { "type": "interim", "title": "Rewrite in the form of a first order separable ODE", "input": "y^{\\prime}\\left(x\\right)+2y=7", "result": "\\frac{1}{7-2y}y^{\\prime}\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Standard form of a first order separable ODE:", "secondary": [ "$$N\\left(y\\right){\\cdot}y^{\\prime}\\left(x\\right)=M\\left(x\\right)$$" ] }, { "type": "step", "result": "y^{^{\\prime}}\\left(x\\right)+2y=7" }, { "type": "step", "primary": "Subtract $$2y$$ from both sides", "result": "y^{^{\\prime}}\\left(x\\right)+2y-2y=7-2y" }, { "type": "step", "primary": "Simplify", "result": "y^{^{\\prime}}\\left(x\\right)=7-2y" }, { "type": "step", "primary": "Divide both sides by $$7-2y$$", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{7-2y}=\\frac{7-2y}{7-2y}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{y^{^{\\prime}}\\left(x\\right)}{7-2y}=1" }, { "type": "step", "primary": "Rewrite in standard form", "secondary": [ "$$N\\left(y\\right)=\\frac{1}{7-2y},\\:{\\quad}M\\left(x\\right)=1$$" ], "result": "\\frac{1}{7-2y}y^{^{\\prime}}\\left(x\\right)=1" } ], "meta": { "interimType": "Canon First Order Separable ODE 2Eq" } }, { "type": "interim", "title": "Solve $$\\frac{1}{7-2y}y^{\\prime}\\left(x\\right)=1:{\\quad}-\\frac{1}{2}\\ln\\left(7-2y\\right)=x+c_{1}$$", "input": "\\frac{1}{7-2y}y^{\\prime}\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "If$${\\quad}N\\left(y\\right)\\cdot\\:y'=M\\left(x\\right),\\:y'=\\frac{dy}{dx},\\:$$then $$\\int{N\\left(y\\right)}dy=\\int{M\\left(x\\right)}dx$$, up to a constant", "result": "\\int\\:\\frac{1}{7-2y}dy=\\int\\:1dx" }, { "type": "step", "primary": "Integrate each side of the equation" }, { "type": "interim", "title": "$$\\int\\:1dx=x+c_{1}$$", "input": "\\int\\:1dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=x+c_{1}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{7-2y}dy=-\\frac{1}{2}\\ln\\left(7-2y\\right)+c_{2}$$", "input": "\\int\\:\\frac{1}{7-2y}dy", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{7-2y}dy", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=7-2y$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dy}=-2$$", "input": "\\left(7-2y\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=7^{^{\\prime}}-\\left(2y\\right)^{^{\\prime}}" }, { "type": "interim", "title": "$$7^{\\prime}=0$$", "input": "7^{\\prime}", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wNdSf6tDSLgQsMQ2V9mr2KboRT4ICWDw+vXUOyCoK0ujkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhL0YItdA45d1Dj9POWnebOU=" } }, { "type": "interim", "title": "$$\\left(2y\\right)^{\\prime}=2$$", "input": "\\left(2y\\right)^{\\prime}", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2y^{^{\\prime}}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$y^{\\prime}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OOqI3g/jkISGM3IRXEvSiMPQlVFV646ejpUuEWqujX2QuIxj9K+Upo9l4tAcwN/gSLdINoPD2MyLmUXT+YnlMnnETkp5DD7YYNXdTglcD6gQDuUeqnBwgbzjs2dJUq2K" } }, { "type": "step", "result": "=0-2" }, { "type": "step", "primary": "Simplify", "result": "=-2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-2dy$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dy=\\left(-\\frac{1}{2}\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}\\left(-\\frac{1}{2}\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{u}\\left(-\\frac{1}{2}\\right):{\\quad}-\\frac{1}{2u}$$", "input": "\\frac{1}{u}\\left(-\\frac{1}{2}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{u}\\cdot\\:\\frac{1}{2}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{1\\cdot\\:1}{u\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=-\\frac{1}{2u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{2u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/qhTB21jGGuAb+ARMcPPl97yAoDmrUJoc9O7IwaMhpg7CmuVIP0/DMlFt6wE9n+ApCJDtyZjhH1bmxVOkEAjY+AYoeqpKSjYJE4Knoiu3x9H8+8dZn2eLZdGMMVBu7+kYEFMST8lDZxn1Yq5HMKVTsGLeMxVl55xMYxEfjKBatulcQUlMOhkqQvF9O8Q8/Z5g==" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{2u}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral:$$\\quad\\:\\int\\:\\frac{1}{u}du=\\ln\\left(u\\right),\\:$$assuming a complex-valued logarithm", "result": "=-\\frac{1}{2}\\ln\\left(u\\right)" }, { "type": "step", "primary": "Substitute back $$u=7-2y$$", "result": "=-\\frac{1}{2}\\ln\\left(7-2y\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{2}\\ln\\left(7-2y\\right)+c_{2}", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "-\\frac{1}{2}\\ln\\left(7-2y\\right)+c_{2}=x+c_{1}" }, { "type": "step", "primary": "Combine the constants", "result": "-\\frac{1}{2}\\ln\\left(7-2y\\right)=x+c_{1}" } ], "meta": { "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Apply initial conditions:$${\\quad}-\\frac{1}{2}\\ln\\left(7-2y\\right)=x-\\frac{\\ln\\left(5\\right)}{2}$$", "input": "-\\frac{1}{2}\\ln\\left(7-2y\\right)=x+c_{1}", "steps": [ { "type": "step", "primary": "Plug in $$x=0$$:$${\\quad}-\\frac{1}{2}\\ln\\left(7-2y\\left(0\\right)\\right)=0+c_{1}{\\quad}$$, and use initial condition $$y\\left(0\\right)=1$$", "result": "-\\frac{1}{2}\\ln\\left(7-2\\cdot\\:1\\right)=0+c_{1}" }, { "type": "interim", "title": "Isolate $$c_{1}:{\\quad}c_{1}=-\\frac{\\ln\\left(5\\right)}{2}$$", "input": "-\\frac{1}{2}\\ln\\left(7-2\\cdot\\:1\\right)=0+c_{1}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "0+c_{1}=-\\frac{1}{2}\\ln\\left(7-2\\cdot\\:1\\right)" }, { "type": "step", "primary": "$$0+c_{1}=c_{1}$$", "result": "c_{1}=-\\frac{1}{2}\\ln\\left(7-2\\cdot\\:1\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "c_{1}=-\\frac{1\\cdot\\:\\ln\\left(7-2\\cdot\\:1\\right)}{2}" }, { "type": "interim", "title": "$$1\\cdot\\:\\ln\\left(7-2\\cdot\\:1\\right)=\\ln\\left(5\\right)$$", "input": "1\\cdot\\:\\ln\\left(7-2\\cdot\\:1\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=1\\cdot\\:\\ln\\left(7-2\\right)" }, { "type": "step", "primary": "Subtract the numbers: $$7-2=5$$", "result": "=1\\cdot\\:\\ln\\left(5\\right)" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\ln\\left(5\\right)=\\ln\\left(5\\right)$$", "result": "=\\ln\\left(5\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BsmQe587Ty/YrLI/v+s1KnIZ3ANpIp64ZY6KKY9u4Uzdd47a0hQ8flDbGsI5To1dkofS+CPlAra4Mq6VfKDcsIxRfhq4bXv9lKhZM7D7OKdqJZwgnCr6rbEYIfM8faxxj02HJ6kG2E65+t0mv9VomLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "c_{1}=-\\frac{\\ln\\left(5\\right)}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "step", "primary": "For $$y=x+c_{1}{\\quad}$$plug in$${\\quad}c_{1}=-\\frac{\\ln\\left(5\\right)}{2}$$", "result": "-\\frac{1}{2}\\ln\\left(7-2y\\right)=x-\\frac{\\ln\\left(5\\right)}{2}" } ], "meta": { "interimType": "Plug In Initial Condition 0Eq" } }, { "type": "interim", "title": "Isolate $$y:{\\quad}y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}$$", "input": "-\\frac{1}{2}\\ln\\left(7-2y\\right)=x-\\frac{\\ln\\left(5\\right)}{2}", "steps": [ { "type": "interim", "title": "Multiply both sides by $$-2$$", "input": "-\\frac{1}{2}\\ln\\left(7-2y\\right)=x-\\frac{\\ln\\left(5\\right)}{2}", "result": "\\ln\\left(7-2y\\right)=-2x+\\ln\\left(5\\right)", "steps": [ { "type": "step", "primary": "Multiply both sides by $$-2$$", "result": "\\left(-\\frac{1}{2}\\ln\\left(7-2y\\right)\\right)\\left(-2\\right)=x\\left(-2\\right)-\\frac{\\ln\\left(5\\right)}{2}\\left(-2\\right)" }, { "type": "interim", "title": "Simplify", "input": "\\left(-\\frac{1}{2}\\ln\\left(7-2y\\right)\\right)\\left(-2\\right)=x\\left(-2\\right)-\\frac{\\ln\\left(5\\right)}{2}\\left(-2\\right)", "result": "\\ln\\left(7-2y\\right)=-2x+\\ln\\left(5\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-\\frac{1}{2}\\ln\\left(7-2y\\right)\\right)\\left(-2\\right):{\\quad}\\ln\\left(7-2y\\right)$$", "input": "\\left(-\\frac{1}{2}\\ln\\left(7-2y\\right)\\right)\\left(-2\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1}{2}\\ln\\left(7-2y\\right)\\cdot\\:2" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}\\ln\\left(7-2y\\right)" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\ln\\left(7-2y\\right)\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$\\ln\\left(7-2y\\right)\\cdot\\:1=\\ln\\left(7-2y\\right)$$", "result": "=\\ln\\left(7-2y\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DPqTQEO9IJ0HThO5ILSNF54ww/bIQkqpoWLI/dGijjItOtZYwUjyXhDTsNnn6ElrEWKbcJL5hTDKZvnQoIKbrFjv4aVLBBsNWKNR3QY0Z2ceNvb7k0sVmuwf19w9aD9NpcoiRSaULsQmcej0DeR3vOZ/VTZAecdLOY2uNTlwj7mh3N5AFcHP+j6+wwgrJI0Z" } }, { "type": "interim", "title": "Simplify $$x\\left(-2\\right)-\\frac{\\ln\\left(5\\right)}{2}\\left(-2\\right):{\\quad}-2x+\\ln\\left(5\\right)$$", "input": "x\\left(-2\\right)-\\frac{\\ln\\left(5\\right)}{2}\\left(-2\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=-x\\cdot\\:2+\\frac{\\ln\\left(5\\right)}{2}\\cdot\\:2" }, { "type": "interim", "title": "$$\\frac{\\ln\\left(5\\right)}{2}\\cdot\\:2=\\ln\\left(5\\right)$$", "input": "\\frac{\\ln\\left(5\\right)}{2}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\ln\\left(5\\right)\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\ln\\left(5\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nQTWPqpV98BiAJ0zaAOYsIjVaQixQyeRDNBeq7Sbb7wgJ/ZZA32ZInFBpDtxBfiKB3icl8DTPbZ2e90WKoHnbj/L0MoYg+CUn6oyL3EO7YoDFwCzBMsc+Dwhs6CjIRD5po5/ngPG1wq/ZpXligh9Zdr8aWu+9GJOampu5Ym1T30=" } }, { "type": "step", "result": "=-2x+\\ln\\left(5\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HTX+VSO5mch8VdN849UWYk6ta/IHbi3pejjuP8ODoURV00rpv8+ZC6TM10tVCSHs160R/f8+selqqEoWafnCuijBHnSmTnR+WDOraXYkY1IeNvb7k0sVmuwf19w9aD9NZJGOBlB5hGDb5PPM2jzxx86L+C57n9Z6sZySI/NcC4+eCZvu43XhmmSOF5PkK9+b" } }, { "type": "step", "result": "\\ln\\left(7-2y\\right)=-2x+\\ln\\left(5\\right)" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply log rules", "input": "\\ln\\left(7-2y\\right)=-2x+\\ln\\left(5\\right)", "result": "7-2y=5e^{-2x}", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\ln\\left(7-2y\\right)=-2x+\\ln\\left(5\\right)\\quad\\:\\Rightarrow\\:\\quad\\:7-2y=e^{-2x+\\ln\\left(5\\right)}$$" ], "result": "7-2y=e^{-2x+\\ln\\left(5\\right)}" }, { "type": "interim", "title": "Expand $$e^{-2x+\\ln\\left(5\\right)}:{\\quad}5e^{-2x}$$", "input": "e^{-2x+\\ln\\left(5\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "result": "=e^{\\ln\\left(5\\right)}e^{-2x}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$e^{\\ln\\left(5\\right)}:{\\quad}5$$", "input": "e^{\\ln\\left(5\\right)}", "result": "=5e^{-2x}", "steps": [ { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "result": "=5", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rRtRaJAvXJWzfoYTsYOjSrU4O2qd+vCZaVi4wMPq8/mbwLLf/4qFQM8wXvPHcutMOsi0ZMrmx6dIX+R0AW4q1EUqTd96MWTKI6Kr2Ib0iQDan2z/3kXxOb0ZWofMdlwEVa/prb+oE2OI1/k9x84pqw==" } }, { "type": "step", "result": "7-2y=5e^{-2x}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Hdr+JNEHluUMhBQ/apyzvhxXAu7JTcd4dhK7bDT/5x8aM37KF/jK0q16Pg+4JBw3XVzMugffw5yS5udvsVvYI6wO9HEPEtO/T1wLTkM8YPB6oAeLYpXJa0ylhWP+jbX3RSpN33oxZMojoqvYhvSJAK4kKVVWunDKeKCY/K9ABdeFqbRXneCsWHKGNxAiWnqv" } }, { "type": "interim", "title": "Solve $$7-2y=5e^{-2x}:{\\quad}y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}$$", "input": "7-2y=5e^{-2x}", "result": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}", "steps": [ { "type": "interim", "title": "Move $$7\\:$$to the right side", "input": "7-2y=5e^{-2x}", "result": "-2y=5e^{-2x}-7", "steps": [ { "type": "step", "primary": "Subtract $$7$$ from both sides", "result": "7-2y-7=5e^{-2x}-7" }, { "type": "step", "primary": "Simplify", "result": "-2y=5e^{-2x}-7" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2y=5e^{-2x}-7", "result": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2y}{-2}=\\frac{5e^{-2x}}{-2}-\\frac{7}{-2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-2y}{-2}=\\frac{5e^{-2x}}{-2}-\\frac{7}{-2}", "result": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-2y}{-2}:{\\quad}y$$", "input": "\\frac{-2y}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2y}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=y" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XUddB5bm3pN+C0CF1rBRQ3yRHuGw7+tM5METTDj6vVH+raPieCBNRuc2kvkIDJQqZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2OlV3GSTqpR1MiV5eh8qI8ialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{5e^{-2x}}{-2}-\\frac{7}{-2}:{\\quad}-\\frac{5e^{-2x}}{2}+\\frac{7}{2}$$", "input": "\\frac{5e^{-2x}}{-2}-\\frac{7}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{5e^{-2x}}{2}-\\frac{7}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{5e^{-2x}}{2}-\\left(-\\frac{7}{2}\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/Mnv/bypkKDbs7ef9semXFqslGDqA0pXUPXGmTK6XqvBxVWi9gPeLj0EZpUKE9UMzMFYmi1F5Hg/ibpEToVnY1o/qN7N0Y8+lgTyaIkhbNPBlmjiUyu2fMqK8bVeiw0iEc7ShOedm97LMngC0LVkYx4pgUWEah0lniZLlD4X0wsb270YgxOvL9l4hXMPkr8ygkYSfPSaiENJU29zdQ3kMwdhF6oJDkPC3RqtunWAo5I=" } }, { "type": "step", "result": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UVeLZRxgfhj7xF9CMkuNnoLzQJqRjKdHPHeUnfKLe91Odta0v9i0ue5RTdWUc9s5BBWuAll+cNN0HgMZQtjoIaGAkgwVNsi7Q6QgVO/F6NNaP6jezdGPPpYE8miJIWzTwZZo4lMrtnzKivG1XosNIgoD9mk5n/r3JH625KfoGBnSFjMXq/GNJmgxIicJj889Bs73n/bKJk6uHxV3H84c+0vSsy16gXkUS+7wd48eQcUyRfq7+ADTi9HigHjUO0rE/WYpGjrCqZBwgmsZkywQ+KfpZTvJtSVLB7MWEsq+Qo1W+6pShiQqYErp9JHZw2WCR+p9PcdLGV2Y23qtRx+/JILKsA73MxANnKMSJYzuojkc0tqYRLKJeQRU1TgF7fJeVfes/snfzU4It1/sLtZZEO9sGZu5A1MXROmEpnxG69pNsGJJ7Fz/BvPOMCZw4Q7gYBEwWupTU1wR9IXQf2HteA==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 1Eq" } }, { "type": "interim", "title": "Check solutions by applying initial conditions", "steps": [ { "type": "step", "primary": "$$y\\left(0\\right)=1$$", "secondary": [ "$$y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}:\\quad\\:1=-\\frac{5e^{-2\\cdot\\:0}}{2}+\\frac{7}{2}$$:$${\\quad}$$True" ] }, { "type": "step", "primary": "Therefore, the final solution for $$y^{\\prime}\\left(x\\right)+2y=7,\\:{\\quad}y\\left(0\\right)=1$$ is " }, { "type": "step", "result": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}" } ], "meta": { "interimType": "Check Solutions Initial Conditions 1Eq" } }, { "type": "step", "result": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}" } ], "meta": { "interimType": "ODE Solve Separable 0Eq" } }, { "type": "step", "result": "y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}" } ], "meta": { "solvingClass": "ODE" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "#>#ODE#>#y=-\\frac{5e^{-2x}}{2}+\\frac{7}{2}" } } }, "meta": { "showVerify": true } }